To find the solutions for the quadratic equation x² + 2x + 3 = 0, we can graph it by converting it into a system of equations. A helpful approach is to express the equation in terms of y:
Equation 1: To represent the quadratic function, we can write:
y = x² + 2x + 3
This equation is a parabola that opens upwards. To find the solutions (or roots) of the original equation, we are essentially looking for the x values where y = 0.
Equation 2: We can create a horizontal line:
y = 0
Now, we have a system of equations:
- y = x² + 2x + 3
- y = 0
When we graph these two equations on the same set of axes, the points where the parabola intersects the horizontal line represent the solutions to the equation x² + 2x + 3 = 0.
However, in this particular case, you might notice an important aspect: the quadratic expression x² + 2x + 3 does not cross the x-axis because the discriminant (b² – 4ac) is negative.
The discriminant can be calculated as follows:
- a = 1 (coefficient of x²)
- b = 2 (coefficient of x)
- c = 3 (constant term)
Calculating the discriminant:
b² – 4ac = 2² – 4(1)(3) = 4 – 12 = -8
Since the discriminant is negative, this indicates that the parabola does not intersect the x-axis, meaning there are no real solutions for the equation x² + 2x + 3 = 0. The solutions are instead complex numbers. The quadratic will have roots:
x = -1 ± i √2
In conclusion, the system of equations to graph would be:
- y = x² + 2x + 3 (the parabola)
- y = 0 (the x-axis)
But it’s crucial to remember that there are no real intersections, indicating no real solutions for the quadratic equation.